A mass m = 4 kg is attached to both a spring with spring constant k = 197 N/m and a dash-pot with damping constant c = 4 N-s/m The mass is started in motion with initial position To = 2 m and initial velocity v0 = 8 m/s. Determine the position function ä(t) in meters. r(t) = Note that, in this problem, the motion of the spring is underdamped, therefore the solution can be written in the form z(t) = C₁e-cos(w₁ta₁). Determine C₁w₁a₁and p. C₁ W1 01 P Graph the function z(t) together with the "amplitude envelope" curves z = C₁e-Pt and x = C₁e-pt Now assume the mass is set in motion with the same initial position and velocity, but with the dashpot disconnected (so c = 0). Solve the resulting differential equation to find the position function u(t) In this case the position function u(t) can be written as u(t) = Cocos (wotao). Determine Co, wo and co. Co= Po= (assume 0 a₁ < 2π) 0 (assume 002π) inally, graph both function (t) and u(t) in the same window to illustrate the effect of damping.

A mass m = 4 kg is attached to both a spring with spring constant k = 197 N/m and a dash-pot with damping constant c = 4 N-s/m The mass is started in motion with initial position To = 2 m and initial velocity v0 = 8 m/s. Determine the position function ä(t) in meters. r(t) = Note that, in this problem, the motion of the spring is underdamped, therefore the solution can be written in the form z(t) = C₁e-cos(w₁ta₁). Determine C₁w₁a₁and p. C₁ W1 01 P Graph the function z(t) together with the "amplitude envelope" curves z = C₁e-Pt and x = C₁e-pt Now assume the mass is set in motion with the same initial position and velocity, but with the dashpot disconnected (so c = 0). Solve the resulting differential equation to find the position function u(t) In this case the position function u(t) can be written as u(t) = Cocos (wotao). Determine Co, wo and co. Co= Po= (assume 0 a₁ < 2π) 0 (assume 002π) inally, graph both function (t) and u(t) in the same window to illustrate the effect of damping.

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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Concept explainers

Free Damped Vibrations

Free vibration is a type of vibration in which the external force is removed after giving the initial displacement to a body/system. In other words, in this type of vibration, force is applied at the initial displacement of the system, and after the initial displacement, force is removed, and the system vibrates at the natural frequency.

Question

A mass m = 4 kg is attached to both a spring with spring constant k = 197 N/m and a dash-pot with damping constant c = 4 N-s/m.
The mass is started in motion with initial position To = 2 m and initial velocity vo = 8 m/s
Determine the position function z(t) in meters.
r(t) =
Note that, in this problem, the motion of the spring is underdamped, therefore the solution can be written in the form z(t) = C₁e-pt cos(w₁ta₁). Determine
C₁,w₁a₁ and p.
C₁₂ =
W₁ =
-α₁ =
P=
Graph the function z(t) together with the "amplitude envelope" curves * = C₁e-Pt and x = C₁e-pt
Now assume the mass is set in motion with the same initial position and velocity, but with the dashpot disconnected (so c = 0) Solve the resulting differential
equation to find the position function u(t).
n this case the position function u(t) can be written as u(t) = Cocos(wotao). Determine Co, wo and co
Co
Po
(assume 0 a₁ < 2π)
20
(assume 000 < 2π)
inally graph both function (t) and u(t) in the same window to illustrate the effect of damping.

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